JERUSALEM — American mediators held urgent contacts with Israeli and Palestinian officials Sunday in hopes of salvaging troubled Mideast peace talks — searching for a formula to bring the sides back together and extend the negotiations beyond a current late-April deadline.

Officials from all sides said diplomacy has picked up over the past 24 hours and an Israeli official, speaking on condition of anonymity because he was not authorized to brief journalists, said talks with the Palestinians via the Americans were held throughout the day.

With the sides unable to agree on the terms of a promised Israeli prisoner release, the negotiations appear to face a risk of collapse in coming weeks. U.S. Secretary of State John Kerry, in Paris, was weighing a return to the region to find a way out of the deadlock.

The issue of the prisoner release “could be resolved or it could fall apart” in the next few days, Israeli Prime Minister Benjamin Netanyahu told a closed meeting of officials from his Likud Party, according to meeting participants. “In any case, there won’t be any deal without getting something in return.” The participants spoke on condition of anonymity because the talks were closed to the media.

Under heavy pressure from Kerry, Israel and the Palestinians agreed in July to hold nine months of peace talks, setting a late-April deadline for a final agreement. When that became unrealistic, Kerry scaled back his goals and said he would aim for a preliminary “framework” agreement by April, with the goal of continuing negotiations through the end of the year to iron out the final details of a deal.

But even that more modest goal has run into trouble due to a snag over the prisoner release. When the talks began in the summer, Israel promised to free 104 long-serving Palestinian prisoners in four stages, with the final release coming by the end of March. After carrying out the first three releases, Israel has balked at releasing the final group without a Palestinian commitment to extend talks.

Israel’s defense minister said Sunday that “this is a critical week for the Israeli-Palestinian issue.” Moshe Yaalon was speaking after meeting visiting U.S. military chief Gen. Martin Dempsey. Yaalon said Israel appreciates the U.S.’s “commitment and contribution” in facilitating talks. Israeli officials say they are under no obligation to carry out the final release because of what they say is a Palestinian failure to negotiate in good faith.

Yuval Steinitz, a Likud Cabinet minister, said “it is clear” the release can’t be carried out if Palestinian President Mahmoud Abbas plans on walking out of the talks the next day. “This release was meant to be carried out as the talks proceed, and not when they fall apart.”

Mohammed Ishtayeh, an aide to Abbas, accused Israel of “trying to blackmail us.”

But Palestinian officials stopped short of threatening to walk out of the talks altogether. The chief Palestinian negotiator, Saeb Erekat, confirmed that negotiations were ongoing and said Abbas was “making every possible effort” to ensure the release of the fourth group.

Palestinian officials said they would drive a tough bargain in exchange for extending the talks. The officials said Abbas would seek the release of 1,000 additional prisoners held by Israel as well as a freeze in Israeli settlement construction in captured territories sought by the Palestinians.

The issue of Palestinian prisoners is deeply emotional on both sides. The Palestinians consider the roughly 5,000 Palestinians held by Israel to be heroes and freedom fighters. Israel considers them to be terrorists. The prisoners released in the previous three tranches had all served lengthy terms for involvement in bloody attacks on Israelis, and scenes of them returning to jubilant celebrations have angered the Israeli public.

Netanyahu said he would present any additional release to his Cabinet — where approval is not guaranteed. Netanyahu’s coalition is dominated by hard-liners who have been extremely critical of the previous releases. The final release is especially contentious because it is expected to include convicted murderers and Arab citizens of Israel.

Kerry was in Paris to meet with his Russian counterpart for talks on the crisis in Ukraine. American officials said Kerry was considering an unscheduled return to the Middle East on Monday in search of a formula to extend the negotiations.

After brokering eight months of negotiations, Kerry has little to show for his efforts. There have been no visible signs of progress, and Israel and the Palestinians have accused each other of hindering progress.

The Palestinians seek the West Bank, east Jerusalem and Gaza Strip — territories captured by Israel in the 1967 Mideast war — for an independent state. They have demanded that the pre-1967 lines serve as the basis of a final border, allowing for small modifications through agreed-upon land swaps to allow Israel to keep some of the settlements it has built.

Netanyahu has refused to accept the 1967 lines as the basis for a final border, and has demanded that Israel maintain a security presence in large parts of the West Bank after any deal is reached. He also rejects any division of east Jerusalem, the Palestinians would-be capital, which is home to sensitive holy sites.

The Palestinians, meanwhile, have rejected Netanyahu’s demand that they recognize Israel as the Jewish homeland. Netanyahu says such acceptance would show the Palestinians are serious about peace. The Palestinians say this demand would harm the rights of Palestinian refugees and Israel’s Arab minority.

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